Which of the following is the length of the unknown leg of a right triangle that has one leg length of 5 feet and a hypotenuse of 17 feet? (Round to the nearest tenth.)
A.
18 feet
B.
14.4 feet
C.
9.8 feet
D.
16.2 feet
The correct answer and Explanation is :
To determine the length of the unknown leg in a right triangle where one leg is 5 feet and the hypotenuse is 17 feet, you can use the Pythagorean theorem. The theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.
Let’s denote the unknown leg as ( x ). According to the Pythagorean theorem:
[ a^2 + b^2 = c^2 ]
where ( a ) and ( b ) are the lengths of the legs of the triangle, and ( c ) is the length of the hypotenuse. In this case:
- One leg ( a = 5 ) feet
- The hypotenuse ( c = 17 ) feet
- The other leg ( b = x )
Substitute these values into the Pythagorean theorem formula:
[ 5^2 + x^2 = 17^2 ]
Calculate ( 5^2 ) and ( 17^2 ):
[ 5^2 = 25 ]
[ 17^2 = 289 ]
So the equation becomes:
[ 25 + x^2 = 289 ]
To solve for ( x^2 ), subtract 25 from both sides:
[ x^2 = 289 – 25 ]
[ x^2 = 264 ]
Next, find ( x ) by taking the square root of 264:
[ x = \sqrt{264} \approx 16.2 ]
Thus, the length of the unknown leg is approximately 16.2 feet. Rounding to the nearest tenth, the correct answer is:
D. 16.2 feet